The compression theorem III: applications

Colin Rourke, Brian Sanderson

Abstract.
This is the third of three papers about the Compression Theorem: if
M^m is embedded in Q^q X R with a normal vector field and if q-m > 0,
then the given vector field can be straightened (ie, made parallel to
the given R direction) by an isotopy of M and normal field in Q x
R. The theorem can be deduced from Gromov's theorem on directed
embeddings [Partial differential relations, Springer-Verlag (1986);
2.4.5 C'] and the first two parts gave proofs. Here we are concerned
with applications. We give short new (and constructive) proofs for
immersion theory and for the loops-suspension theorem of James et al
and a new approach to classifying embeddings of manifolds in
codimension one or more, which leads to theoretical solutions. We also
consider the general problem of controlling the singularities of a
smooth projection up to C^0-small isotopy and give a theoretical
solution in the codimension >0 case.